Gelfand pairs and strong transitivity for Euclidean buildings

Let G be a locally compact group acting properly by type-preserving automorphisms on a locally finite thick Euclidean building $Δ$ and K be the stabilizer of a special vertex in $Δ$. It is known that (G, K) is a Gelfand pair as soon as G acts strongly transitively on $Δ$; this is in particular the case when G is a semi-simple algebraic group over a local field. We show a converse to this statement, namely: if (G, K) is a Gelfand pair and G acts cocompactly on $Δ$, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in G and their peculiar dynamics on the spherical building at infinity. Other equivalent formulations are also obtained, including the fact that G is strongly transitive on $Δ$ if and only if it is strongly transitive on the spherical building at infinity.

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